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arXiv · 1711.02562

The topological entropy of endomorphisms of Lie groups

Mauro Patrão
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Abstract

In this paper, we determine the topological entropy $h(ϕ)$ of a continuous endomorphism $ϕ$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(ϕ)$ associated to a left invariant distance $d$ just provides an upper bound to $h(ϕ)$, which is characterized by the so called variational principle. We prove that \[ h\left(ϕ\right) = h\left(ϕ|_{T(G_ϕ)}\right) \] where $G_ϕ$ is the maximal connected subgroup of $G$ such that $ϕ(G_ϕ) = G_ϕ$, and $T(G_ϕ)$ is the maximal torus in the center of $G_ϕ$. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.

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Mauro Patrão. 2018-04-29. The topological entropy of endomorphisms of Lie groups. https://arxiv.org/abs/1711.02562

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